PHYSICAL REVIEW X 8, 031042 (2018) Landau Effective Interaction between Quasiparticles in a Bose-Einstein Condensate A. Camacho-Guardian* and Georg M. Bruun Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark (Received 19 December 2017; revised manuscript received 28 February 2018; published 15 August 2018) Landau’s description of the excitations in a macroscopic system in terms of quasiparticles stands out as one of the highlights in quantum physics. It provides an accurate description of otherwise prohibitively complex many-body systems and has led to the development of several key technologies. In this paper, we investigate theoretically the Landau effective interaction between quasiparticles, so-called Bose polarons, formed by impurity particles immersed in a Bose-Einstein condensate (BEC). In the limit of weak interactions between the impurities and the BEC, we derive rigorous results for the effective interaction. They show that it can be strong even for a weak impurity-boson interaction, if the transferred momentum- energy between the quasiparticles is resonant with a sound mode in the BEC. We then develop a diagrammatic scheme to calculate the effective interaction for arbitrary coupling strengths, which recovers the correct weak-coupling results. Using this scheme, we show that the Landau effective interaction, in general, is significantly stronger than that between quasiparticles in a Fermi gas, mainly because a BEC is more compressible than a Fermi gas. The interaction is particularly large near the unitarity limit of the impurity-boson scattering or when the quasiparticle momentum is close to the threshold for momentum relaxation in the BEC. Finally, we show how the Landau effective interaction leads to a sizable shift of the quasiparticle energy with an increasing impurity concentration, which should be detectable with present- day experimental techniques. DOI: 10.1103/PhysRevX.8.031042 Subject Areas: Atomic and Molecular Physics I. INTRODUCTION interaction is caused by the exchange of gauge bosons in particle physics [6]. Landau’s quasiparticle theory represents a powerful The experimental realization of highly population- framework for making accurate predictions about quantum imbalanced atomic gases, where the minority (impurity) many-body systems [1,2]. The quasiparticle concept dra- atoms form quasiparticles, has significantly advanced our matically reduces the complexity of the problem, and it is understanding of this fundamental topic. In particular, since exploited across many areas of physics. The quasiparticle the interaction between the impurity atoms and the majority framework has led to technological breakthroughs such as atoms can be tuned using Feshbach resonances [7], one can semiconductor devices and superconductors, and it is study quasiparticle physics systematically and in regimes central for the development of future quantum technolo- never realized before. Impurity atoms were first realized gies. The effective interaction between quasiparticles plays ’ experimentally in degenerate Fermi gases, where they form a key role in Landau s theory, where it determines both – thermodynamic and dynamical properties. Interactions quasiparticles called Fermi polarons [8 11].Wenowhavea good understanding of the Fermi polaron, even for strong between quasiparticles are observed in liquid-helium mix- – tures [3], and they are the origin of important effects such as interactions between the impurity and the Fermi gas [12 19]. Recently, two experiments observed long-lived quasiparticles conventional and high-Tc superconductivity, where the interaction is mediated by lattice vibrations and spin formed by impurity atoms in a Bose-Einstein condensate fluctuations, respectively [4,5]. Likewise, the fundamental [20,21], following an earlier experiment on impurities in a one-dimensional (1D) Bose gas [22]. These quasiparticles, called Bose polarons, have been analyzed theoretically using a wide range of techniques [23–35]. *Corresponding author. Most of the theoretical and experimental studies in [email protected] atomic gases focus on the properties of a single polaron, and much less is known about the interaction between Published by the American Physical Society under the terms of polarons. In light of its key importance, a natural question is the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to then whether the great flexibility of atomic-gas experiments the author(s) and the published article’s title, journal citation, can be used to obtain new insights into Landau’s effective and DOI. interaction, like it did concerning the properties of a single 2160-3308=18=8(3)=031042(12) 031042-1 Published by the American Physical Society A. CAMACHO-GUARDIAN and GEORG M. BRUUN PHYS. REV. X 8, 031042 (2018) polaron. Studies of the Landau effective interaction have assume that the bosons are weakly interacting with a gas been limited to vanishing momenta using perturbation 1=3 ≪ 1 0 parameter nB aB , where aB > is the boson-boson theory [36], and using variational and diagrammatic tech- scattering length. The normal and anomalous BEC Green’s niques for Fermi polarons [11,14,37], or to the case of two functions for the bosons are, respectively, fixed impurities in real space using perturbation theory and 2 2 variational techniques [38–40]. uk vk G11ðk;zÞ¼ − ; In this paper, we systematically investigate the Landau z − Ek z þ Ek effective interaction between Bose polarons, both as a ukvk ukvk G12ðk;zÞ¼ − ; ð2Þ function of momenta and as a function of the boson- z þ Ek z − Ek impurity interaction strength. We derive rigorous results ¼½ϵBðϵB þ 2μ Þ1=2 for a weak interaction using perturbation theory. It is where Ek k k B is the Bogoliubov spectrum, μ ¼ ð ÞT shown that, even in this limit, the interaction between B n0 T B is the chemical potential of the bosons, and 2 2 ¼½ðϵB þ μ Þ Æ 1 2 two polarons can be strong when the momentum-energy uk=vk k B =Ek = are the usual coherence T ¼ 4π exchange is resonant with a sound mode in the Bose- factors. We define B aB=mB. Einstein condensate (BEC). Our perturbative calculation is The impurities interact with the BEC, forming quasipar- then generalized to arbitrary interaction strengths using a ticles denoted Bose polarons or simply polarons when there diagrammatic resummation scheme. We show that the is no ambiguity. To avoid confusion, we remind the reader Landau effective interaction, in general, is much stronger that Bose polarons refer to mobile impurities in a Bose gas. than that between Fermi polarons. The interaction is The impurities themselves can be either bosonic or fermionic particularly strong when the boson-impurity interaction as will indeed be discussed in detail below. A polaron is close to the unitarity limit or when the momentum of one with momentum p is described by the Green’s function −1 −1 −1 of the polarons approaches the onset of momentum Gðp;zÞ ¼ G0ðp;zÞ −Σðp;zÞ, where G0ðp;zÞ ¼ relaxation caused by scattering bosons out of the BEC. z−ξp is the noninteracting Green’sfunctionandΣðp;zÞ is The strong effective interaction is then demonstrated to give the self-energy. The polaron energy εp is found by solving rise to a substantial shift in the polaron energy as a function ε ¼ ξ þ Σðp ε Þ ð Þ of the impurity concentration. We conclude by discussing p p Re ; p ; 3 how such effects can be observed experimentally. where we assume that the imaginary part of the self-energy is ¼½1 − ∂Σðp Þ ∂ −1j small and the residue Zp ;z = z z¼εp of the II. BOSE POLARONS quasiparticle is close to unity so that the polaron is well We consider mobile impurities of mass m, either bosonic defined. In the following, we for simplicity focus on the or fermionic, immersed in a gas of bosons of mass m . The case of an attractive interaction between the impurity and the B BEC corresponding to a negative boson-impurity scattering Hamiltonian of the system is 0 Σðp Þ X X length a< . We calculate the self-energy ;z using the B † ˆ gB † † finite-temperature field theory. H ¼ ϵ b bk þ b b 0 bk0 bk k k 2V kþq k −q k k;k0q X X † g † † III. LANDAU EFFECTIVE INTERACTION þ ξkc ck þ b c 0 ck0 bk; ð1Þ k V kþq k −q k k;k0q Even though there is no direct interaction between the impu- rities, two polarons interact via their effects on the surrounding † † where bk and ck create a boson and an impurity, respec- BEC or, equivalently, due to the exchange of Bogoliubov B tively, with momentum k, and V is the volume. Here, ϵk ¼ sound modes. Landau’s effective interaction between two 2 p p k =2mB is the kinetic energy of the bosons, and ξk ¼ polarons with momenta 1 and 2 is defined as [3] k2=2m − μ is the kinetic energy of the impurity atoms δεp δReΣðp1; εp Þ μ ðp p Þ¼ 1 ¼ 1 ð Þ relative to their chemical potential . The boson-boson and f 1; 2 Zp1 Zp1 ; 4 δnp δnp boson-fermion interactions are taken to be short range with 2 2 strengths gB and g, and we assume that there is no direct where np is the quasiparticle distribution function and we use interaction between the impurities. This assumption is due Eq. (3) in the second equality. to the Pauli principle for fermionic impurities, whereas any direct interaction between bosonic impurities can easily be A. Weak coupling included at the mean-field level. We first consider the weakly interacting regime Below the critical temperature Tc, the bosons form a j j ≪ 1 ¼ð6π2 Þ1=3 BEC with condensate density n0ðTÞ. The total density of kn a , where kn nB is the momentum scale set by the BEC. This regime allows us to derive analytical the bosons is nB, and the density n of the impurities is taken 1=3 to be so small that the bosons are unaffected.